Why Gravitational PE is Usually Negligible for Charges

Good morning. In a previous lesson, we solved for the final speed of a charge when it starts at rest and moves through a uniform electric field from point A to point B. I'm not going to walk through that math because we did it before. However, when solving that problem, we stated that the change in gravitational potential energy of the system was negligible. Today, we are going to prove that to be true. — Flipics, [singing] — right? I thought the change in gravitational potential energy being negligible would mean that the mass of the particle would be negligible. However, charges in the equation for the final speed of the charged particle. H how can the change in gravitational potential energy be negligible and the mass of the charge affect the final speed of the charge? How can both of those be true? — Absolutely. That is the question we are answering today. — Oh, yay.

Okay, let's add in the gravitational field in red. To make things simpler for us, I have oriented the parallel plates parallel to the surface of the Earth. This makes it so the electric field and gravitational field are in the same direction, which makes this much easier. Okay, let's put some numbers in this example to help us understand that the change in gravitational potential energy in this example is in fact negligible. Let's say the charge is a proton and a proton has a mass of 1. 67 67 * 10 -27 kg and a charge of positive 1. 60 * 10 -19 khms. A common electric potential difference between two plates could be 12 vol which is negative when going from point A to point B. And let's say the distance between the two PL plates is roughly 1 mm or 0. 001 m which is on the order of what you could expect for parallel charge plates like this. That means the change in vertical position of the proton would be 0. 001 m because the charge moved down in the direction of the gravitational field. Bobby, determine the change in electric potential energy of the charge plate systems after the charge has moved from point A to point B. Bo, determine the change in gravitational potential energy of the charge plates earth system after the charge has moved from point A to point B. and Billy determine the ratio of those two changes

ratio of the ratio of those two changes in potential energy. Sure. Change in electric potential energy equals charge time electric potential difference or 1. 60 * 10 -19 * -12 or 1. 92 * 10 -18 Jew. And change in gravitational potential energy in a uniform gravitational field equals mass time gravitational field strength time change in vertical position or 1. 67 * 10 uh what is it 27 * 9. 81 *01 or -1. 638 638 * 10 to the -29 JW. And the change in electric potential energy over the change in gravitational potential energy equals 1. 92 * 10 -18 JW over 1. 63A * 10 the 29 JW or uh 1. 171 * 10 11 or roughly 1. 2 2 * 10 11 and it has no units because jewels cancel out. In other words, a rough estimate is that the change in electric potential energy of the system is 100 billion times larger than the change in gravitational potential energy of the system. — Wow. Okay, that is the definition of negligible. Thanks. — Exactly. Because the change in electric potential energy of the system is roughly 100 billion times larger than the change in gravitational potential energy of the system. We can ignore the change in gravitational potential energy and it will have negligible effect on our results. However, the mass of the charge is still in our equation for the speed of the charge at point B because mass is a measure of inertia or a measure of an object's resistance to acceleration. So the larger the mass of the charge, the smaller the acceleration of the charge and the smaller the final speed of the charge at point B. In other words, the mass of the charge affects the acceleration of the charge. However, the mass of the charge has negligible effect on the change in mechanical energy of the charge plates earth system. Thank you very much for learning with me today. I enjoy learning with you.